Compressibility of Air in Casting Mould Evaluated by Finite Element Method and the Stream Function Model

: Casting is a manufacturing process for making complex shapes of metal materials. Casting has two stages – filling process requiring a gating system and solidification process requiring a riser. The air in the mould cavity during casting is displaced through the riser by the molten metal in a very rapid manner necessitating the need to determine its compressibility as it exit through the riser. The finite element method and the stream function model were used to analyze the flow of air through the top risers of casting mould. Results show that the velocity profile at any cross section is parabolic in shape with the maximum velocity at the centre. Comparing the finite element solutions with the exact solutions showed that the solutions converged towards the exact solutions. Further comparing of finite element solutions and experimental results with the local speed of sound (Mach number) showed that the Mach number was greater than one, which established that the air in the mould cavity during casting is compressible as it is displaced by the molten metal. Before now all researches did was to develop empirical equations and optimized molten metal flow in casting mould. This work has gone further to establish the compressibility of airflow in casting mould. permits unrestricted use, distribution, and reproduction in any medium, provided the original is properly

Casting a manufacturing process for making complex shapes of metal materials, have two main stagesfilling process and solidification process (Feng, 2008).The filling process uses the gating system which composed of pouring cup, runner, sprue, sprue-well and ingate, is designed to guide liquid metal into the mould cavity.The solidification process uses the riser system to compensate for shrinkage caused by casting solidification (Feng, 2008).Apart from serving as a reservoir to compensate for shrinkage during solidification, the riser also serve as a channel through which the air displaced as a result of filling the mould cavity with the molten metal goes out of the mould cavity (Inegbedion and Akpobi, 2017a).
For the manufacturing engineer there are many situations where compressible flow understanding is essential for adequate design.These processes include situations not expected to have a compressible flow, such as casting and injection moulding (Meir, 2013).Casting is a process in which liquid metal is injected into a mould to obtain a near final shape.The air is displaced by the liquid metal in a very rapid manner, in a matter of milliseconds; therefore its compressibility has to be taken into account (Meir, 2013).
Porosity the most persistent and common complaint of casting users contributes directly to customer's concern about reliability and quality (Monroe, 2005).To control porosity an understanding of its source and causes is essential.One source of porosity in casting is a failure to eliminate all the air in the Mould Cavity during mould filling (Scott and Goodman, 1978).Aqida et al., (2004) examined the effects of porosity on mechanical properties of metal matrix composite and observed that porosity tends to decrease the mechanical properties of metal composite.Peti and Grama, (2011) described porosity as trapped air in the casting which can come from several sources.Much work has been done to characterize the factors that causes porosity in casting, and to analyze the impact of porosity on the mechanical properties of metals (Aqida et al., 2004).What has not yet been done is to analyze the behaviour of air (the main cause of porosity) in casting mould.
The aim of this work therefore was to determine the compressibility of air in casting mould as liquid metal is poured to fill the mould cavity using the finite element method and the stream function model.(Reddy, 1993, 2006and Singiresu, 2004).The model equation is the stream function model of axisymmetric flow of air in casting mould (Inegbedion and Akpobi, 2017b),

MATERIALS AND METHODS
The weighted residual of equation ( 1) is We integrated equation ( 2) over the element domain Ωe (3) (5) We integrated equation ( 5) by parts with respect to z and r using the basic relation The finite element model of equation ( 7) is given in matrix form as equation ( 8) The stream function model over the domain of interest is discretized into finite elements having M nodes, using suitable interpolation model for ψ (e) in element e as (Reddy, 2006): The interpolation functions ( ) z r N i , are the same as those developed for linear rectangular elements, with x = r and y = z (Inegbedion and Akpobi, 2017b).This will enable us to evaluate the integrals of the e ij K and e ij f .Let's consider an approximation of the form: and using a rectangular element with sides a and b (Figure 1a).
We substituted equation ( 16) into equation ( 14) and noting that a = r and b = z, we obtained ( ) We differentiated equations ( 18) with respect to r and z We rewrote [K e ] in equation ( 9) as the sum of four basic matrices and using the interpolation function of ( 19) evaluated the several e K of each matrix using Figures 1a & 1b ( ) Similarly we evaluated all elements of Similarly we evaluated all elements of Similarly we evaluated all elements of K 4 matrix Equation ( 20 Typical riser height is twice riser diameter (H=2D) for top risers opened to atmospheric pressure (Flinn 1963).The height (z) to diameter (d) ratio used in this work is 2:1.Therefore, b=z=2mm and a=r=0.5mmand equation ( 29 Next we evaluated the e f matrix using equation (11)   The velocity entering the boundary ab is U1 = 1588.6288mm/s(Inegbedion and Akpobi, 2017a) and hence the vector e f will be nonzero only for elements 1 and 3 (Figure 1b).These nonzero vectors can be computed as follows (Singiresu, 2004): Where Ωji denote the lengths of the edge z We assembled the system matrix using Figure 1b and equation ( 8) Finally we evaluated the system matrix  8500 -59.4082 -1054.4221 191.2748 -305.0951 -2403.6911 608.6242 898.4816 -568.3382

RESULTS AND DISCUSSION
Applying the boundary conditions with respect to Figure 2 we obtained the following results as shown in Table 1:  The velocity profile at any cross section is parabolic in shape with the maximum velocity at the centre (Figure 3).The finite element solutions converged towards the exact solution (Figure 4).The finite element results when compared with the experimental results (Inegbedion and Akpobi, 2017a) and the Mach number showed that the air flow in the casting mould was compressible.

Conclusion:
In this work we have used the finite element method and the stream function model to establish the compressibility of air flow in casting mould.Results showed that the velocity profile at any

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INEGBEDION, F; AKPOBI, JA cross section of the riser is parabolic in shape with the maximum velocity at the centre.Comparing results with exact solution shows that the finite element solution converged towards the exact solution.The finite element results were also compared with the experimental results obtained from the measurement of air velocity from the riser using a rotary vane anemometer.

Fig 1a :
Fig 1a: Geometry of the element

Fig 2 :
Fig 2: Computational domain and boundary conditions for the stream-function formulation.

Fig 3 :
Fig 3: Graph of velocity against nodal values showing the Velocity profile at different cross section along the riser Compressibility of Air in Casting Mould…..

Table 1 :
Finite Element solutions from the Analysis of Axisymmetric Flow of Air through the top riser of Casting using the Stream Function Model

Nodes r (mm) z (mm) Stream function
Fig 4: graph of velocity against nodal values showing the Velocity profile at different cross section of the riser this work and Exact solutions